2 research outputs found
Well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge
For a supersonic Euler flow past a straight wedge whose vertex angle is less
than the extreme angle, there exists a shock-front emanating from the wedge
vertex, and the shock-front is usually strong especially when the vertex angle
of the wedge is large. In this paper, we establish the well-posedness for
two-dimensional steady supersonic Euler flows past a Lipschitz wedge whose
boundary slope function has small total variation, when the total variation of
the incoming flow is sufficiently small. In this case, the Lipschitz wedge
perturbs the flow and the waves reflect after interacting with the strong
shock-front or the wedge boundary. We first obtain the existence of solutions
in when the incoming flow has small total variation by the wave front
tracking method and then study the stability of the solutions. In
particular, we incorporate the nonlinear waves generated from the wedge
boundary to develop a Lyapunov functional between two solutions, which is
equivalent to the norm, and prove that the functional decreases in the
flow direction. Then the stability is established, so is the uniqueness
of the solutions by the wave front tracking method. Finally, we show the
uniqueness of solutions in a broader class, i.e. the class of viscosity
solutions.Comment: 29 pages, 1 figur
-Stability of Vortex Sheets and Entropy Waves in Steady Compressible Supersonic Euler Flows over Lipschitz Walls
We study the well-posedness of compressible vortex sheets and entropy waves
in two-dimensional steady supersonic Euler flows over Lipschitz walls with
incoming flows. Both the Lipschitz wall of tangential angle function and
the incoming flow perturb a background strong vortex sheet/entropy wave.
In particular, when the total variation of the incoming flow perturbation
around the background strong vortex sheet/entropy wave is small, we prove that
the two-dimensional steady supersonic Euler flows containing a strong vortex
sheet/entropy wave past the Lipschitz wall are --stable. The weak waves
are reflected after the nonlinear waves interact with the strong vortex
sheet/entropy wave and the wall boundary. Using the wave-front tracking method,
the existence of solutions in over the Lipschitz walls is first shown,
when the total variation of the incoming flow perturbation around the
background strong vortex sheet/entropy wave is suitably small. Then we
establish the --stability of the solutions with respect to the incoming
flows. To achieve this, a Lyapunov functional, equivalent to the
--distance between two solutions containing the strong vortex
sheets/entropy waves, is carefully constructed to include the nonlinear waves
generated by both the wall boundary and the incoming flow. This Lyapunov
functional is then proved to decrease in the flow direction, leading to the
--stability of the solutions. Furthermore, the uniqueness of these
solutions extends to a larger class of viscosity solutions.Comment: 27 pages, 1 figur